Prüfer v-multiplication domains and related domains of the form D+DS[Γ*]

Let D be an integral domain, S be a saturated multiplicative subset of D with D{subset of with not equal to}DS, and Γ be a nonzero torsion-free grading monoid with Γ∩-Γ={0}. Let DS[Γ] be the semigroup ring of Γ over D_S, Γ*=Γ-{0}, and D^(S,Γ)=D+D_S[Γ*], i.e., D^(S,Γ)={f∈D_S[Γ]|f(0)∈D}. We show that D^(S,Γ) is a P. vMD (resp., GCD-domain, GGCD-domain) if and only if D is a P. vMD (resp., GCD-domain, GGCD-domain), Γ is a valuation semigroup and S is a t-splitting (resp., splitting, d-splitting) set of D.